Question:
Show that $f(x)=x+\cos x-a$ is an increasing function on $R$ for all values of a ?
Solution:
We have,
$f(x)=x+\cos x-a$
$f^{\prime}(x)=1-\sin x=\frac{2 \cos ^{2} x}{2}$
Now,
$x \in R$
$\Rightarrow \frac{\cos ^{2} x}{2}>0$
$\Rightarrow \frac{2 \cos ^{2} x}{2}>0$
$\Rightarrow f^{\prime}(x)>0$
Hence, $f(x)$ is an increasing function for $x \in R$
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